Digital audio is recorded using digital sampling techniques at standard sample rates of 32 KHz, 44.1 KHz, or 48 KHz with maximum recordable frequencies of 16 KHz, 22.050 KHz, and 24 KHz, respectively. These limitations are known as Nyquist Criterion.
Playback of digital audio recreates the recorded audio data correctly up to the maximum frequency or Nyquist rate. In addition, an audio data image is also created. In a frequency spectrum, this image is the reverse of the original audio information offset by half the sample frequency, see FIG. 3. An increasing audio frequency signal generates decreasing frequency signal above half the sampling frequency i.e. a 20 KHz tone will create an image tone at
44.1 KHz-20 KHz=24.1 KHz, PA1 44.1 KHz-10 KHz=33.1 KHz. PA1 88.2 KHz-22.05 KHz=66.15 KHz. PA1 22.05 KHz-20 KHz=2.5 KHz. PA1 88.2 KHz-22.05 KHz=66.16 KHz,
a 10 KHz tone will create an image tone at:
The audio data image produced in playback must be removed in order to achieve acceptable playback of the original audio information. With the audio data image directly above the original audio data information, an analog filter with high filtering capability is needed to remove the audio data image. The audio data image can be shifted in frequency to a frequency span high above the original audio information to relax analog image removal filter requirements.
The audio data image is shifted in frequency by use of an unsampling interpolation digital filter or multirate filter. The multirate filter performs interpolation between data point of the original audio data by increasing the sampling rate. The increased sampling rate allows for filtering above the original Nyquist rate.
The multirate filter is required to have a cutoff at the Nyquist rate and maximum attenuation at the lowest possible audio data image. Audio material recorded at 44.1 KHz will have a maximum recorded frequency of 22.05 KHz and a minimum possible audio data image of 22.05 KHz. An image removal filter is required to have a passband of 20 KHz and a stopband of 22.05 KHz, with an attenuation in the stopband greater than 90 dB. Use of a digital filter with these specifications and an ending sampling rate of two time the original sampling rate (2 Fs) would create a possible minimum audio data image at
The audio data image could be shifted to even higher frequencies of 4 Fs, 8 Fs, 16 Fs, 32 Fs, etc using additional multirate filtering.
It is however necessary to use multirate filtering at least at twice the original sample rate, which is the most critical and demanding filter. At 2 Fs the transition band is
A 4 FS filter which is required to have a 20 KHz passband must filter the audio data image at
and has a transition band of 46.15 KHz. The 2 Fs filter is a factor of 12 worse than the 4 Fs filter. Higher filter rates have even larger transition bands. The extremely narrow transition band of the 2 Fs digital filter highlights the Gibbs phenomena.
The Gibbs Phenomena is the ringing effect which can be observed on transient responses of digital filters due to a finite window of points used to perform the multirate filtering. Gibbs phenomena creates a ringing distortion before and after a sharp transient. FIGS. 1a and 2a illustrate the effect of the ringing on a short positive going pulse and on a square wave signal. The pre-ringing distortion is highly audible in music passages and is the major audio difference between analog processed playback and digital playback.